Sunday, November 23, 2014

`int x tan^2 x dx` Evaluate the integral

`intxtan^2xdx`


 Rewrite the integrand using the identity `tan^2x=sec^2x-1`


`intxtan^2xdx=intx(sec^2x-1)dx`


`=intxsec^2xdx-intxdx`


Now let's evaluate `intxsec^2xdx`  using integration by parts,


`intxsec^2xdx=x*intsec^2xdx-int(d/dx(x)intsec^2(x))dx`


`=xtan(x)-int(1*tan(x))dx`


`=xtan(x)-int(sin(x)/cos(x))dx`


Substitute cos(x)=t


-sin(x)dx=dt


`int(sin(x)/cos(x))dx=int-dt/t`


`=-ln|t|`


substitute back t=cos(x),


`=-ln|cos(x)|`


`intxsec^2xdx=xtan(x)-(-ln|cos(x)|)`


`=xtan(x)+ln|cos(x)|`


`intxtan^2(x)dx=xtan(x)+ln|cos(x)|-intxdx`


`=xtan(x)+ln|cos(x)|-x^2/2+C`


C is a constant

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